AP Calculus BC · Unit 9 Parametric/Vector Calculus
Learning Objectives
Decoupled Axes: In parametric equations, the $x$ and $y$ positions are completely independent functions of a hidden third variable, Time $t$.
Vector Velocity: Since $x$ and $y$ move separately, velocity is a vector $\langle \frac{dx}{dt}, \frac{dy}{dt} \rangle$. The absolute speed is the magnitude $\sqrt{(x')^2 + (y')^2}$.
The Derivative $\frac{dy}{dx}$: By the Chain Rule, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. This gives the geometric slope of the trace path, completely ignoring *how fast* the particle was moving!
Tags
ParametricVectorsVelocitySpeedChain Rule
PARAMETRIC SYSTEM $x(t)$ = t - \sin(t) $y(t)$ = 1 - \cos(t)
Select Parametric Path
Time Control (t)
Current Time $t$ = 0.00
Instantaneous Vectors
AXIAL VELOCITY (COMPONENT)
$\frac{dx}{dt} =$0.00
$\frac{dy}{dt} =$0.00
VECTOR SYNTHESIS
Speed $|v| =$0.00
Slope $\frac{dy}{dx} =$0.00
Cycloid Analysis: The path traced by a point on a rolling wheel. At $t=0, 2\pi$, both $\frac{dx}{dt}$ and $\frac{dy}{dt}$ hit $0$. Notice how the particle literally stops moving entirely at the cusps on the ground!